Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  459.10002
Autor:  Erdös, Paul
Title:  Problems and results in number theory. (In English)
Source:  Recent progress in analytic number theory, Symp. Durham 1979, Vol. 1, 1- 13 (1981).
Review:  [For the entire collection see Zbl 451.00001.]
In the author's words, ``somewhat unconventional problems on sieves, primes and congruences'' are discussed. Of the wealth of the problems, let us take a few examples. The integer n is said to be a barriere for an arithmetic function f if m+f(m) \leq n for all m < n. Question: are there infinitely many barriers for \epsilon\upsilon(n), for some \epsilon > 0? Here \upsilon(n) denotes the number of distinct prime factors of n. A related problem: is it true that limn ––> oomaxm < n(m+d(m))-n = oo? There are also many problems concerning consecutive primes. Let dn = pn+1-pn. To prove that for any c > 0 there are infinitely many k such that

dk > c log k log log k log log log log k/(log log log k)2

(10000 dollars offered for a proof).
Reviewer:  M.Jutila
Classif.:  * 11-02 Research monographs (number theory)
                   11Axx Elementary number theory
                   11Mxx Analytic theory of zeta and L-functions
                   00A07 Problem books
Keywords:  problems in number theory; sieves; primes; congruences; barrier for arithmetic function; consecutive primes
Citations:  Zbl.451.00001

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